My question is about using prescaling for Heron's Algorithm as described in on page 4 in this textbook: http://assets.press.princeton.edu/chapters/s9487.pdf
I am able to understand that we are limiting our solution only to nonnegative numbers since we are searching for real roots. But I do not follow why this textbook has chosen the interval of $[\frac{1}{2}, 2]$ and the corresponding transformation of $$\tilde{y} = 4^k y$$ if $y\not\in[\frac{1}{2},2]$ for some integer $k$. Both the transformation and the interval appear arbitrarily chosen and I am wondering how to generalize and understand the scaling for other values and since the remainder of the chapter seems to require understanding this transformation.
Heron's algorithm converges quadratically, but only if your start value is not far away from the true root (in relative terms). By applying the algorithm to numbers $y$ in the interval $\bigl[{1\over2},2]$ and starting with $x_0=1$ we make sure that ${x_0/\sqrt{y}}$ is pretty close to $1$.
If the start value $x_0$ is (relatively) far away from the true root then during the first steps of the algorithm the relative error is only halved (and not squared) at each step (see page $6$ of the text). The suggested prescaling (which is for free in binary) allows to start right away with a "sensible" approximation to the true root.